Symplectic Structures and Geometry of Canonical Bundle

正则丛的辛结构和几何

• 批准号: 10440021
• 负责人: MABUCHI Toshiki
• 金额: $ 7.42万
• 依托单位: Osaka University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B).
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-10440021/
• 关键词: symplactic structure; Kahler structure; canonical bundle; anticanonical bundle; Kahler-Einstein metric; Hitchin-Kobayashi correspondence; Futaki choracter; stability; シンプレティック構造; Tian; Kahler-Einstein軽量; 複数幾何; 複素幾何

We focussed our study on the "Hitchin-Kobayashi correspondence for manifolds" which is one of the most interesting topics in our project. For a Fano manifold, such a correspondence is supposed to relate the Chow-Mumford stability of the manifold with the existence of Kahler-Einstein metrics. As a first step, we obtained :(1) For a certain generalization of Kahler-Einstein metrics, we showed the uniqueness of such metrics modulo holomorphic automorphisms on a given Fano manifold. Hence, even in this generalized context, the above correspondence is one-to-one. (In the case of Kahler-Ricci solitons, a similar result was obtained also by Tian and Zhu.)Recently, we also saw the following :(2) In a joint work with H.Nakagawa, we succeeded in characterizing the Futaki character of a Fano manifold as an obstruction to Chow-Mumford semistability of the manifold.(3) In complex analytic studies of a polarized manifold, the asymptotic stability of the manifold has a strong relationship, via the asymptotic behavior of the Bergman metrics, with the existence of metrics of constant scalar curvature in the polarized Kahler class.

我们把我们的研究集中在“流形的Hitchin-Kobayashi对应”上，这是我们项目中最有趣的话题之一。对于一个Fano流形，这样的对应关系被认为是将流形的Chow-Mumford稳定性与Kahler-Einstein度量的存在性联系起来。作为第一步，我们得到：（1）对于Kahler-Einstein度量的一个推广，我们证明了在给定的Fano流形上这类度量模全纯自同构的唯一性。因此，即使在这种广义的上下文中，上述对应关系也是一对一的。(In对于Kahler-Ricci孤子，Tian和Zhu也得到了类似的结果。（2）在与H.Nakagawa的合作中，我们成功地刻画了Fano流形的Futaki特征标，它是流形的Chow-Mumford半稳定性的障碍。(3)在极化流形的复分析研究中，流形的渐近稳定性通过Bergman度量的渐近行为与极化Kahler类中常数量曲率度量的存在性有很强的关系。

项目成果

Kobayahi, R.: "Holomorphic curves in abelian varieties : the second main theorem and applications"Japanese J.Math.. Vol.26. 129-152 (2000)

Kobayahi, R.：“阿贝尔簇中的全纯曲线：第二个主要定理和应用”Japan J.Math.. Vol.26。

E.Klimenko, M.Sakuma: "Two-generator discrete subgroups of Isom(H^2)containing orientation-reversing elements" Geom. Dedicata. 72. 247-282 (1998)

E.Klimenko，M.Sakuma：“包含方向反转元素的 Isom(H^2) 的两个生成离散子群”Geom。

Konno,K.: "clifford index and the slope of fibered surfaces"J.Algebraic Geom.. 8. 207-220 (1999)

Konno,K.：“克利福德指数和纤维表面的斜率”J.Algebraic Geom.. 8. 207-220 (1999)

Mabuchi,T.: "Vector field energies and critical metrics on Kahler manifolds"Nagoya Math.J. (to appear). 162. (2001)

Mabuchi,T.：“卡勒流形上的矢量场能量和关键度量”Nagoya Math.J.

Mabuchi,T.: "Kahler-Einstein metrics for manifolds with nonvanishing Futaki character"Tohoku Math.J. (to appear). 53. (2001)

Mabuchi,T.：“具有非零二木特征的流形的卡勒-爱因斯坦度量”Tohoku Math.J.